Induced charge on a plane

The question itself is easy. E field at 2 is zero. E field at other points are of magnitude [itex]\frac{\eta}{2\epsilon}[/itex]

But I am arguing with my classmate and tutor that what is the charge induced on the uncharged conductor.

For me, I simply use the rule induced charge (charge of same magnitude but different sign will be induced) to conclude that the charge density on the bottom of the conductor must be [itex]-\eta[/itex].
Since it is uncharged, the charge density on the top must be [itex]\eta[/itex].

But my classmate use Gauss law and obtain that the charge density on the bottom of the conductor is [itex]\frac{-\eta}{2}[/itex].

My tutor also said that the answer provided by professor is [itex]\frac{-\eta}{2}[/itex].

If the charge density on the top of the uncharged conductor was η, the field in region 1 would be η/ε which is not correct.
The rule of induced charge is applicable to grounded conductors. Here the uncharged conductor is isolated.